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In this article, an event-triggering impulsive control strategy is proposed, in which the impulsive time sequence can be different from the event-triggered time sequence. The synchronization problems of a multiple neural network with delay (MDNN) and the directed disconnected switching topology are discussed by using this strategy. The considered switching topology is assumed to be directed and sequentially or jointly connected. The combined measurement method is adopted in the event-triggering strategy, so that each delayed neural network only updates the control rules at the moment of its event triggering. First, we prove that the designed event-triggering rules can avoid Zeno behavior. Then, the sufficient conditions of event-based quasi-synchronization (synchronization) for the MDNN can be obtained by using the iterative method. In addition, as an extension, the case of a jointly connected topology and the case of a pure impulsive control protocol are considered, too. Finally, a numerical example is provided to test the results in theory analysis. [ABSTRACT FROM AUTHOR]